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This 9 page Class Notes was uploaded by Zoekgaber on Wednesday March 2, 2016. The Class Notes belongs to PHYS 1311-11 at Tulane University taught by Dean MacLaren in Spring 2016. Since its upload, it has received 13 views. For similar materials see Physics I (Calc Based) in Physics 2 at Tulane University.
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Date Created: 03/02/16
Lab 2: Kinematics Members: Samantha Rubin, Marguerite Furlong, Zoe Gaber Table Number: 2 Introduction Kinematics explores the relationship between objects and motion, particularly for this lab, an object in free fall. Velocity and acceleration are mathematically represented with their derivatives in respect to either position or velocity in terms of time: Here, Δx is identified as the initial position of x subtracted from the final position of x, and Δv, is the initial velocity subtracted from the final velocity. When acceleration is constant with respect to time, we can relate the following kinematic equations to define the relationships between position, velocity, and acceleration: Here, V r0fers to the initial velocity and X 0efers to the initial position, both in terms of time. The known acceleration is that of gravity, or the gravitational pull of the earth, which is defined as 9.8m/s . In this lab we examine the quantity of g in the real life data collected from an object in free fall. Experiment 1 Measuring the Acceleration by Gravity In this experiment a plastic rectangle with alternating clear and black horizontal stripes was dropped straight down, passing through a Photogate sensor. The plastic rectangle, referred to as a picket fence, measured 39cm and each black or clear stripe measured 2.5cm. The fence was dropped from a height of 72cm over a number of trials. The fence passed through a Photogate sensor mounted on a metal pole 70 cm above the table where the pole was attached. The Photogate sensor has two plastic prongs that stick out horizontally parallel to each other with an infrared beam of light running between them. As the picket fence falls between the two prongs, the infrared light is able to detect when it is block by each of the black stripes passing through. From this, the data can be transferred to the LoggerPro computer program, where it is used to calculate how quickly the picket fence was falling by measuring the amount of time it took to get from one black stripe to the next, a distance of 5cm. Because the height of the picket fence is known and the distance between the bars is known, velocity and acceleration can be calculated. A total of six trials were run and recorded dropping the picket fence past the Photogate sensor to confirm the accuracy of the velocity and acceleration plots. Additional trials were run, but resulted in unusable data due to lack of recording by the Photogate sensor or the picket fence not being dropped straight down and hitting the side of the sensor. Data Analysis Figure 1 Figure 2 Data Trial No. 1 2 3 4 5 6 Slope 9.650 9.796 9.427 9.782 9.450 9.168 (m/s^2) Both the position versus time graph and the velocity versus time graph are linear with positive slopes. The position versus time graph has a much steeper slope (figure 1), looking to be at least double what the slope is for velocity versus time (figure 2), which makes sense because the velocity over time graph is a derivative of the position versus time graph. Of the six trials run the minimum acceleration was 9.168 and the maximum acceleration was 9.796. The average acceleration for the six trials was 9.545m/s 2. The uncertainty of the experiment was 0.314. Final experimental acceleration is 9.545±0.314, or a range of 9.231 to 9.859. Uncertainty is 3.29% of the acceleration, or our experiment had a precision of 3.29%. The generally accepted value of g is 9.8m/s, which falls within our range of the average experimental acceleration plus or minus the uncertainty, although none of our recorded accelerations were 9.8 or above. Additional Analysis The plastic picket fence did not experiment much drag force during its fall due to its small size and thickness. If the picket fence had experienced more drag force as a result of air resistance, it would have had more forces resisting it as it fell down, or countering the force of gravity. Air resistance, or drag force, will increase as a square of an increase in velocity, so we can expect to see acceleration decreasing until it reaches a hypothetical 0. As a result, the graph of velocity vesus time will be upwards and concave, with the slope gradually decreasing until it is a straight line parallel with the x axis. The graph of position versus time will be similar to the graphs recorded in our trials where air resistance was neglected but the slope is expected to be a little smaller due to the effect of drag. Position = y Displacement = y(final)−y(initial)/t(final)−t(initial) or Δy/Δt or d(y)/d(t) Velocity = Δx/Δt or d(r)/d(t) Acceleration = Δv/Δt or d(v)/d(t) Jerk = Δa/Δt or d(a)/d(t) We do not expect to see jerk displayed in this experiment because the acceleration is the constant acceleration of gravity on the picket fence and nothing else, thus the graphs should be linear. Experiment 2: Motion in One Dimension In this experiment, we threw a ball straight upward to determine the changing position of a ball through the use of a ultrasonic motion detector. We then determined the velocity and acceleration graphs of the ball as it is in freefall through the use of the data collection program Logger Pro. We obtained data from tossing a ball above an ultrasonic motion detector, which emits a sound wave that reflects off the ball in motion. The motion detector was put under a metal cage with square bars big enough to not interfere with tracking the sound waves as the ball moves away from the motion detector and back. This sound wave is reflected back to the motion detector and determines the position of the object as it falls by recording the time between when the actual emitted and reflected waves are detected. This information was relayed into the LoggerPro computer system to further analyze the ball’s movement over time, in order to calculate velocity and acceleration. The ball was held 20cm above the metal cage before tossing it upwards to a final height of 60cm. The ball was allowed to fall back down and hit the metal cage holding the motion detector to track its entire movement for the whole bounce, instead of catching it while it was still midair. Care was taken to prevent the experimenter’s hands from interfering with the motion detectors by moving them minimally when tossing the ball and also moving them out of the way of the ball’s path horizontally after the ball had started moving upwards. Only one trial was run resulting in usable data, although previous trials were also conducted but they resulted in discrepancies on the part of the ball due to horizontal movement from the experimenter’s hands or from the ball falling sideways on its decent. From the position data, we were able to determine the velocity vs time and acceleration vs time graph, through the use of Logger Pro (Figure 3). Figure 3 Data Analysis: 1. Figure 4 Figure 4 Analysis: The ball’s maximum velocity was 1.769 m/s, achieved at .6000 seconds. Figure 5 The maximum height of the ball was .5812 meters at .800 seconds The velocity of the ball at the top of its motion was recorded as .211 m/s. (very close to the accepted value of zero) The acceleration when the ball is at the top of its motion (maximum height of the ball) was recorded as 9.296 m/s^2, this is .504 less than the textbook value of g (9.8 m/s^2). 2. A quadratic curve fit to the parabolic portion of our data: Figure 6 y=4.660t^2 + 7.663t + 2.568 The value of (½)g is (½)(9.8) = 4.9; Our coefficient of the x^2 term in the curve is 4.660, which is .24 less than the value of (½)g. The initial velocity coefficient is 7.663 m/s, which is very close to the Yintercept on the Velocity graph of 7.502 m/s. This shows consistency through our data. 3. When the Linear fit was applied to the velocity vs. time graph: Figure 7 Figure 7 Analysis: v = v0 + at v = 7.502 + (9.116)t The coefficient of the x term is 9.116, which is .684 less than the accepted value for g of 9.8. 4. The mean value for the acceleration during the freefall section of flight: Figure 8 The mean value for the acceleration during the freefall section of flight was calculated to be 4.046 m/s^2. The value of g from the position vs. time curve: 9.296 The value of g from the velocity vs time curve: 9.116 m/s^2 Additional Analysis: Position graph for ball thrown with greater starting velocity (more force) Figure 9 Position graph for ball thrown with lighter starting velocity (less force) Figure 10 The parabolic shape of the position graph remains the same, whether the initial velocity is smaller or larger. This shown by figures 9 and 10. However, the time it took to reach the maximum height of the ball was faster when there was a greater starting velocity compared to the graph with a smaller starting velocity, creating a wider parabolic shape. Calculations 2 (9.650+9.796+9.427+9.782+9.450+9.168)/6= 9.5455 m/s 9.7969.168=0.628 0.628/2=0.314 0.314/9.545=0.3289*100=3.29% Conclusion In conclusion, this experiment succeeded in terms of its purpose for both Experiment 1: Measuring the Acceleration by Gravity and Experiment 2: Motion in One Dimension. In the first experiment, we successfully observed the acceleration of gravity in our data. From the graphs of position vs. time and velocity vs. time, the slope, and therefore the acceleration of the picket fence could be recorded. The average acceleration that we recorded was 9.545m/s^2. The accepted value of the acceleration of gravity is 9.8m/s2 and, although most of our recorded accelerations were slightly below 9.8, our acceleration still falls within our range of the average experimental acceleration plus or minus the uncertainty. The uncertainty in this lab is most likely due the slow processing of the Photogate sensor, not drag force because an impact of air resistance would have given a higher percent uncertainty. Overall, the Experiment 1 was successful in calculating a relative acceleration quantity to the accepted number quantified as the gravitational pull at Earth’s surface, or 9.8m/s^2. Observing a range of data close to this quantity proves that our trials were successful. In Experiment 2, we were successfully able to analyze the relationships between the position, velocity and acceleration graphs with respect to time when the object is in free fall. Using our initial data and presenting a Position vs. Time graph in Logger Pro (Figure 5), we were able to further derive a Velocity vs. Time graph by taking the derivative of the Position graph. Further, the derivative of the Velocity function enabled us to create an Acceleration function vs. Time. From our graphs, we were able to determine a value of g. For ourg on the Position and Velocity graphs, they were .504 and .684 less than the accepted value ofg (9.8 m/s^2). This could be attributed to the drag force on the ball while it was in freefall and the air resistance acting on the ball.In addition, the motion detector used in our experiment demonstrated a slow sampling rate, which lead to a worse error for the acceleration graph. Also due to the slow sampling rate, every single point on the time scale is not plotted and the Logger Pro program is unable to successfully zoom in to certain moments without skewing the data through the derivative equation. Overall, the lab went fairly well with minimal sources of error and presented data that was understandable and concise. We were able to obtain a position, velocity, and acceleration function of the Ball Toss.
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